How to make mesh grow to create form
Your idea with the curve isn't bad at all, but for the form of the log you need to alter the geometry as well, I would suggest, create the logo as one mesh, create the shape as simple as possible, in this case just use a plane create the outline, cut the angles with the knife-tool and keep it with only quads, it should be very easy to create this mesh ;-)
When you have the finished mesh, create a few shape-keys, where you just move the vertices back to where they should start. Then insert Keyframes to your timeline, to control the different shape-keys. All the best Marco

For a better understanding here is a quick example of how I would do it.
Here is a little Video https://youtu.be/OW1WdF42yrk on how it would work. And here is a dropboxlink to the example file https://www.dropbox.com/s/aas26i1z0t2e5wa/Logo_folded.blend?dl=0

All the best Marco